On Renewal Matrices Connected with Branching Processes with Tails of Distributions of Different Orders

On Renewal Matrices Connected with Branching Processes with Tails of Distributions of Different...
Topchiĭ, V.
2018-05-30 00:00:00
We study irreducible renewal matrices generated by matrices whose rows are proportional to various distribution functions. Such matrices arise in studies of multi-dimensional critical Bellman–Harris branching processes. Proofs of limit theorems for such branching processes are based on asymptotic properties of a chosen family of renewal matrices. In the theory of branching processes, unsolved problems are known that correspond to the case in which the tails of some of the above mentioned distribution functions are integrable, while the other distributions lack this property.We assume that the heaviest tails are regularly varying at the infinity with parameter −β ∈ [−1, 0) and asymptotically proportional, while the other tails are infinitesimal with respect to them. Under a series of additional conditions, we describe asymptotic properties of the first and second order increments for the renewal matrices.
http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.pngSiberian Advances in MathematicsSpringer Journalshttp://www.deepdyve.com/lp/springer-journals/on-renewal-matrices-connected-with-branching-processes-with-tails-of-S2iRGv6zow

On Renewal Matrices Connected with Branching Processes with Tails of Distributions of Different Orders

Abstract

We study irreducible renewal matrices generated by matrices whose rows are proportional to various distribution functions. Such matrices arise in studies of multi-dimensional critical Bellman–Harris branching processes. Proofs of limit theorems for such branching processes are based on asymptotic properties of a chosen family of renewal matrices. In the theory of branching processes, unsolved problems are known that correspond to the case in which the tails of some of the above mentioned distribution functions are integrable, while the other distributions lack this property.We assume that the heaviest tails are regularly varying at the infinity with parameter −β ∈ [−1, 0) and asymptotically proportional, while the other tails are infinitesimal with respect to them. Under a series of additional conditions, we describe asymptotic properties of the first and second order increments for the renewal matrices.

Journal

Siberian Advances in Mathematics
– Springer Journals

Published: May 30, 2018

Recommended Articles

Loading...

References

Branching Processes

Athreya, K. B.; Ney, P. E.

Regular Variation

Bingham, N. H.; Goldie, C. M.; Teugels, J. L.

Probability Theory

Borovkov, A. A.

Strong renewal theorems with infinite mean

Erickson, K. B.

An introduction to Probability Theory and its Applications

Feller, W.

The Theory ofMatrices

Gantmacher, F. R.

Linear Algebra

Il’in, V. A.; Poznyak, E. G.

Banach algebras of absolutely continuous measures on the straight line

Rogozin, B. A.; Sgibnev, M. S.

Branching Processes

Sevast’yanov, B. A.

Banach algebras of measures of class G(γ)

Sgibnev, M. S.

A note on a multi-dimensional renewal equation

Shurenkov, V.M.

Workshop on Branching Processes and Their Applications

Topchiĭ, V.

Derivative of renewal density with infinite moment with α ∈ (0, 1/2]

Topchiĭ, V. A.

The asymptotic behaviour of derivatives of the renewal function for distributions with infinite first moment and regularly varying tails of index β ∈ (1/2, 1]